3.44.96 \(\int \frac {e^{\frac {-3 x+x^2+e^{2 x} (-5 x-2 x^2)+e^{2 x} \log (-3 x+x^2)}{x}} (-3 x^2+x^3+e^{2 x} (-3+2 x+36 x^2-4 x^4)+e^{2 x} (3-7 x+2 x^2) \log (-3 x+x^2))}{-3 x^2+x^3} \, dx\)

Optimal. Leaf size=28 \[ e^{-3+x-e^{2 x} \left (5+2 x-\frac {\log ((-3+x) x)}{x}\right )} \]

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Rubi [F]  time = 18.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {-3 x+x^2+e^{2 x} \left (-5 x-2 x^2\right )+e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (-3 x^2+x^3+e^{2 x} \left (-3+2 x+36 x^2-4 x^4\right )+e^{2 x} \left (3-7 x+2 x^2\right ) \log \left (-3 x+x^2\right )\right )}{-3 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-3*x + x^2 + E^(2*x)*(-5*x - 2*x^2) + E^(2*x)*Log[-3*x + x^2])/x)*(-3*x^2 + x^3 + E^(2*x)*(-3 + 2*x +
36*x^2 - 4*x^4) + E^(2*x)*(3 - 7*x + 2*x^2)*Log[-3*x + x^2]))/(-3*x^2 + x^3),x]

[Out]

Defer[Int][E^(-3 + x + (E^(2*x)*(-5*x - 2*x^2))/x + (E^(2*x)*Log[-3*x + x^2])/x), x] - 12*Defer[Int][E^(-3 + 3
*x + (E^(2*x)*(-5*x - 2*x^2))/x + (E^(2*x)*Log[-3*x + x^2])/x), x] + Defer[Int][E^(-3 + 3*x + (E^(2*x)*(-5*x -
2*x^2))/x + (E^(2*x)*Log[-3*x + x^2])/x)/(-3 + x), x]/3 + Defer[Int][E^(-3 + 3*x + (E^(2*x)*(-5*x - 2*x^2))/x
+ (E^(2*x)*Log[-3*x + x^2])/x)/x^2, x] - Defer[Int][E^(-3 + 3*x + (E^(2*x)*(-5*x - 2*x^2))/x + (E^(2*x)*Log[-
3*x + x^2])/x)/x, x]/3 - 4*Defer[Int][E^(-3 + 3*x + (E^(2*x)*(-5*x - 2*x^2))/x + (E^(2*x)*Log[-3*x + x^2])/x)*
x, x] - Defer[Int][(E^(-3 + 3*x + (E^(2*x)*(-5*x - 2*x^2))/x + (E^(2*x)*Log[-3*x + x^2])/x)*Log[(-3 + x)*x])/x
^2, x] + 2*Defer[Int][(E^(-3 + 3*x + (E^(2*x)*(-5*x - 2*x^2))/x + (E^(2*x)*Log[-3*x + x^2])/x)*Log[(-3 + x)*x]
)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-3 x+x^2+e^{2 x} \left (-5 x-2 x^2\right )+e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (-3 x^2+x^3+e^{2 x} \left (-3+2 x+36 x^2-4 x^4\right )+e^{2 x} \left (3-7 x+2 x^2\right ) \log \left (-3 x+x^2\right )\right )}{(-3+x) x^2} \, dx\\ &=\int \frac {\exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (3 x^2-x^3-e^{2 x} \left (-3+2 x+36 x^2-4 x^4\right )-e^{2 x} \left (3-7 x+2 x^2\right ) \log \left (-3 x+x^2\right )\right )}{(3-x) x^2} \, dx\\ &=\int \left (\exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )-\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (3-2 x-36 x^2+4 x^4-3 \log ((-3+x) x)+7 x \log ((-3+x) x)-2 x^2 \log ((-3+x) x)\right )}{(-3+x) x^2}\right ) \, dx\\ &=\int \exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \, dx-\int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (3-2 x-36 x^2+4 x^4-3 \log ((-3+x) x)+7 x \log ((-3+x) x)-2 x^2 \log ((-3+x) x)\right )}{(-3+x) x^2} \, dx\\ &=\int \exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \, dx-\int \left (\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (3-2 x-36 x^2+4 x^4\right )}{(-3+x) x^2}-\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) (-1+2 x) \log ((-3+x) x)}{x^2}\right ) \, dx\\ &=\int \exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \, dx-\int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \left (3-2 x-36 x^2+4 x^4\right )}{(-3+x) x^2} \, dx+\int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) (-1+2 x) \log ((-3+x) x)}{x^2} \, dx\\ &=\int \exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \, dx-\int \left (12 \exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )-\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )}{3 (-3+x)}-\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )}{x^2}+\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )}{3 x}+4 \exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) x\right ) \, dx+\int \left (-\frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \log ((-3+x) x)}{x^2}+\frac {2 \exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \log ((-3+x) x)}{x}\right ) \, dx\\ &=\frac {1}{3} \int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )}{-3+x} \, dx-\frac {1}{3} \int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )}{x} \, dx+2 \int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \log ((-3+x) x)}{x} \, dx-4 \int \exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) x \, dx-12 \int \exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \, dx+\int \exp \left (-3+x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \, dx+\int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right )}{x^2} \, dx-\int \frac {\exp \left (-3+3 x+\frac {e^{2 x} \left (-5 x-2 x^2\right )}{x}+\frac {e^{2 x} \log \left (-3 x+x^2\right )}{x}\right ) \log ((-3+x) x)}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 33, normalized size = 1.18 \begin {gather*} e^{-3+x-e^{2 x} (5+2 x)} ((-3+x) x)^{\frac {e^{2 x}}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-3*x + x^2 + E^(2*x)*(-5*x - 2*x^2) + E^(2*x)*Log[-3*x + x^2])/x)*(-3*x^2 + x^3 + E^(2*x)*(-3 +
2*x + 36*x^2 - 4*x^4) + E^(2*x)*(3 - 7*x + 2*x^2)*Log[-3*x + x^2]))/(-3*x^2 + x^3),x]

[Out]

E^(-3 + x - E^(2*x)*(5 + 2*x))*((-3 + x)*x)^(E^(2*x)/x)

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fricas [A]  time = 0.63, size = 40, normalized size = 1.43 \begin {gather*} e^{\left (\frac {x^{2} - {\left (2 \, x^{2} + 5 \, x\right )} e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (x^{2} - 3 \, x\right ) - 3 \, x}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-7*x+3)*exp(x)^2*log(x^2-3*x)+(-4*x^4+36*x^2+2*x-3)*exp(x)^2+x^3-3*x^2)*exp((exp(x)^2*log(x^2
-3*x)+(-2*x^2-5*x)*exp(x)^2+x^2-3*x)/x)/(x^3-3*x^2),x, algorithm="fricas")

[Out]

e^((x^2 - (2*x^2 + 5*x)*e^(2*x) + e^(2*x)*log(x^2 - 3*x) - 3*x)/x)

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giac [A]  time = 0.37, size = 33, normalized size = 1.18 \begin {gather*} e^{\left (-2 \, x e^{\left (2 \, x\right )} + x + \frac {e^{\left (2 \, x\right )} \log \left (x^{2} - 3 \, x\right )}{x} - 5 \, e^{\left (2 \, x\right )} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-7*x+3)*exp(x)^2*log(x^2-3*x)+(-4*x^4+36*x^2+2*x-3)*exp(x)^2+x^3-3*x^2)*exp((exp(x)^2*log(x^2
-3*x)+(-2*x^2-5*x)*exp(x)^2+x^2-3*x)/x)/(x^3-3*x^2),x, algorithm="giac")

[Out]

e^(-2*x*e^(2*x) + x + e^(2*x)*log(x^2 - 3*x)/x - 5*e^(2*x) - 3)

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maple [C]  time = 0.19, size = 149, normalized size = 5.32

method	result	size
risch	\(x^{\frac {{\mathrm e}^{2 x}}{x}} \left (x -3\right )^{\frac {{\mathrm e}^{2 x}}{x}} {\mathrm e}^{-\frac {i {\mathrm e}^{2 x} \pi \mathrm {csgn}\left (i x \left (x -3\right )\right )^{3}-i {\mathrm e}^{2 x} \pi \mathrm {csgn}\left (i x \left (x -3\right )\right )^{2} \mathrm {csgn}\left (i x \right )-i {\mathrm e}^{2 x} \pi \mathrm {csgn}\left (i x \left (x -3\right )\right )^{2} \mathrm {csgn}\left (i \left (x -3\right )\right )+i {\mathrm e}^{2 x} \pi \,\mathrm {csgn}\left (i x \left (x -3\right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -3\right )\right )+4 \,{\mathrm e}^{2 x} x^{2}+10 x \,{\mathrm e}^{2 x}-2 x^{2}+6 x}{2 x}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-7*x+3)*exp(x)^2*ln(x^2-3*x)+(-4*x^4+36*x^2+2*x-3)*exp(x)^2+x^3-3*x^2)*exp((exp(x)^2*ln(x^2-3*x)+(-
2*x^2-5*x)*exp(x)^2+x^2-3*x)/x)/(x^3-3*x^2),x,method=_RETURNVERBOSE)

[Out]

x^(exp(2*x)/x)*(x-3)^(exp(2*x)/x)*exp(-1/2*(I*exp(2*x)*Pi*csgn(I*x*(x-3))^3-I*exp(2*x)*Pi*csgn(I*x*(x-3))^2*cs
gn(I*x)-I*exp(2*x)*Pi*csgn(I*x*(x-3))^2*csgn(I*(x-3))+I*exp(2*x)*Pi*csgn(I*x*(x-3))*csgn(I*x)*csgn(I*(x-3))+4*
exp(2*x)*x^2+10*x*exp(2*x)-2*x^2+6*x)/x)

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maxima [A]  time = 0.54, size = 39, normalized size = 1.39 \begin {gather*} e^{\left (-2 \, x e^{\left (2 \, x\right )} + x + \frac {e^{\left (2 \, x\right )} \log \left (x - 3\right )}{x} + \frac {e^{\left (2 \, x\right )} \log \relax (x)}{x} - 5 \, e^{\left (2 \, x\right )} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-7*x+3)*exp(x)^2*log(x^2-3*x)+(-4*x^4+36*x^2+2*x-3)*exp(x)^2+x^3-3*x^2)*exp((exp(x)^2*log(x^2
-3*x)+(-2*x^2-5*x)*exp(x)^2+x^2-3*x)/x)/(x^3-3*x^2),x, algorithm="maxima")

[Out]

e^(-2*x*e^(2*x) + x + e^(2*x)*log(x - 3)/x + e^(2*x)*log(x)/x - 5*e^(2*x) - 3)

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mupad [B]  time = 3.36, size = 36, normalized size = 1.29 \begin {gather*} {\mathrm {e}}^{-5\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^x\,{\left (x^2-3\,x\right )}^{\frac {{\mathrm {e}}^{2\,x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(3*x + exp(2*x)*(5*x + 2*x^2) - exp(2*x)*log(x^2 - 3*x) - x^2)/x)*(exp(2*x)*(2*x + 36*x^2 - 4*x^4 -
3) - 3*x^2 + x^3 + exp(2*x)*log(x^2 - 3*x)*(2*x^2 - 7*x + 3)))/(3*x^2 - x^3),x)

[Out]

exp(-5*exp(2*x))*exp(-3)*exp(-2*x*exp(2*x))*exp(x)*(x^2 - 3*x)^(exp(2*x)/x)

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sympy [A]  time = 0.96, size = 37, normalized size = 1.32 \begin {gather*} e^{\frac {x^{2} - 3 x + \left (- 2 x^{2} - 5 x\right ) e^{2 x} + e^{2 x} \log {\left (x^{2} - 3 x \right )}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-7*x+3)*exp(x)**2*ln(x**2-3*x)+(-4*x**4+36*x**2+2*x-3)*exp(x)**2+x**3-3*x**2)*exp((exp(x)**2
*ln(x**2-3*x)+(-2*x**2-5*x)*exp(x)**2+x**2-3*x)/x)/(x**3-3*x**2),x)

[Out]

exp((x**2 - 3*x + (-2*x**2 - 5*x)*exp(2*x) + exp(2*x)*log(x**2 - 3*x))/x)

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